#846574

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=fg,d^2=g,e^2=f,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	#72297741, #72297742, #72297773
maximal normal subgroup of	#72297741, #72297742, #72297773
maximal quotient of	#72298563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 753770);