#185102

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2176 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2176 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239885
maximal normal subgroup of	#72239885
maximal quotient of	#72239896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92298);