#875303

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3956 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)(7,9)(8,11)(10,12)(14,18)(16,19)(21,22)(23,24)(25,26)(27,28);(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(22,26)(23,27);(5,6,9,7)(8,10,12,11)(13,15,17,20)(14,19,18,16)(21,23,25,27)(22,24,26,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,17,19)(14,20,18,15)(21,24)(22,27)(23,26)(25,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,27)(23,26)(25,28)

Polycyclic presentation:

2^9|c^2=f,d^2=ghi,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	#72326462, #72326488, #72326528
maximal normal subgroup of	#72326462, #72326488, #72326528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 782499);