#6403912

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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×D₄²	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 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)(7,11)(8,10)(9,12)(13,14,19,22)(15,23,17,20)(16,27,21,18)(24,25,28,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,28)(17,19)(18,26)(21,24)(22,23)(25,27);(6,10)(9,11)(13,16,18,14,19,21,27,22)(15,24,26,23,17,28,25,20);(7,12)(9,11)(13,17,19,15)(14,20,22,23)(16,28,21,24)(18,25,27,26);(1,3,4,2)(6,10)(9,11)(13,18,19,27)(14,21,22,16)(15,25,17,26)(20,28,23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311108);