#7527836

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2259 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256861, #72256888, #72256984, …
maximal normal subgroup of	#72256861, #72256888, #72256984, …
maximal quotient of	#72264703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7435032);