#357319

order	512
nilpotency class	3
isoclinism	isoclinic class of #23360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cef,d^b=de,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264515);