#7611556

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

2143 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)(7,9)(8,10)(11,12)(13,14,18,23)(15,24,26,20)(16,27,25,21)(17,28,19,22);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,19)(17,25)(18,26)(21,22)(23,24)(27,28);(6,10)(7,11)(13,14)(15,24)(16,27)(17,28)(18,23)(19,22)(20,26)(21,25);(6,10)(9,12)(13,16,17,26,18,25,19,15)(14,21,28,24,23,27,22,20);(1,3,4,2)(13,17,18,19)(14,22,23,28)(15,25,26,16)(20,21,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308261, #72308340, #72308754, …
maximal normal subgroup of	#72308261, #72308340, #72308754, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518752);