#7148404

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1954 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)(7,11)(10,12)(13,14)(15,22)(16,19)(17,21)(18,26)(20,24)(23,28)(25,27);(6,9)(7,11)(13,15,19,23)(14,20,16,25)(17,24,18,27)(21,28,26,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,14)(15,20,23,25)(17,21,18,26)(22,27,28,24);(6,9)(10,12)(13,17)(14,21)(16,26)(18,19)(20,25)(24,27);(5,7,8,11)(6,10,9,12)(14,16)(15,23)(21,26)(24,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307895, #72308060, #72308395
maximal normal subgroup of	#72307895, #72308060, #72308395

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055600);