#777502

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2535	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3236 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)(7,11)(8,12)(9,10)(13,14,17,19)(15,20,16,18)(23,27)(26,28);(6,10)(8,11)(13,15)(14,18)(16,17)(19,20)(22,25)(26,28);(13,14)(15,20)(16,18)(17,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,19)(21,23,24,27)(22,26,25,28);(5,8,9,11)(6,12,10,7)(13,16,17,15)(14,18,19,20)

Polycyclic presentation:

2^9|a^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297804, #72297829, #72297841
maximal normal subgroup of	#72297804, #72297829, #72297841
maximal quotient of	#72298302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 684698);