#6373262

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#34911	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2296 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286522
maximal normal subgroup of	#72286522
maximal quotient of	#72286847

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280458);