#7613601

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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

2208 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:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257118, #72257119, #72257163
maximal normal subgroup of	#72257118, #72257119, #72257163
maximal quotient of	#72264651

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520797);