#6389356

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279850, #72280005
maximal normal subgroup of	#72279850, #72280005

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296552);