#7064221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2129 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307330, #72307669, #72307855
maximal normal subgroup of	#72307330, #72307669, #72307855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6971417);