#874102

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

3282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^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	#72333355, #72333388, #72333413
maximal normal subgroup of	#72333355, #72333388, #72333413

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 781298);