#6444973

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307127, #72307285, #72307315
maximal normal subgroup of	#72307127, #72307285, #72307315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352169);