#856230

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328077, #72328136, #72328195, …
maximal normal subgroup of	#72328077, #72328136, #72328195, …
maximal quotient of	#72328618, #72331323

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 763426);