#158798

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

3136 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=gh,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	#72328492, #72328972, #72329202
maximal normal subgroup of	#72328492, #72328972, #72329202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65994);