#7614543

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2109 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)(7,11)(8,12)(9,10)(13,14)(15,23)(16,21)(17,22)(18,26)(19,25)(20,24)(27,28);(2,4)(6,10)(11,12)(13,15,19,16)(14,21,25,23)(17,20,27,18)(22,26,28,24);(5,7,9,8)(6,11,10,12)(13,16,18,17,19,15,20,27)(14,22,24,21,25,28,26,23);(1,2,3,4)(13,17,20,16,19,27,18,15)(14,23,26,28,25,21,24,22);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,17,16,27)(21,22,23,28)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=h,e^2=f,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257074, #72257113, #72257166
maximal normal subgroup of	#72257074, #72257113, #72257166
maximal quotient of	#72265665

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521739);