#7501215

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35563	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2400 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308676, #72308698, #72308854
maximal normal subgroup of	#72308676, #72308698, #72308854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408411);