#6817302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgh,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240756, #72257394, #72264140
maximal normal subgroup of	#72240756, #72257394, #72264140
maximal quotient of	#72240770

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724498);