#156767

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329918, #72330189, #72330273, …
maximal normal subgroup of	#72329918, #72330189, #72330273, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 63963);