#7633414

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

4436 conjugacy classes of subgroups; only the 3170 normal subgroups are shown.

normal subgroups only

This group has 4436 conjugacy classes of subgroups — too many to draw. Only its 3170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26)(27,30)(28,32)(29,31)(33,36)(34,37)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(10,15)(14,20)(16,22)(21,24)(27,33)(30,36)(34,39)(37,40);(11,17)(14,20)(18,23)(21,24)(26,31)(27,33)(32,38)(34,39);(25,28,29,35)(26,32,31,38)(27,34,33,39)(30,37,36,40);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72238834, #72238835
Fitting subgroup of	#72238834
maximal subgroup of	#72209972, #72209974, #72238826, …
maximal normal subgroup of	#72209972, #72209974, #72238826, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540610);