#6606709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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
prime radical	C₂×#3347	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263905, #72263950, #72264017
maximal normal subgroup of	#72263905, #72263950, #72264017
maximal quotient of	#72265115, #72307883

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513905);