#6463233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2741 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224734
maximal normal subgroup of	#72224734
maximal quotient of	#72224720, #72224721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370429);