#6388997

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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)(8,11)(9,12)(13,14,18,23)(15,24,19,20)(16,26,17,22)(21,27,28,25);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15,18,19)(14,20,23,24)(16,25,17,27)(21,26,28,22);(13,16,18,17)(14,21,23,28)(15,25,19,27)(20,26,24,22);(5,8,10,9)(6,11,7,12)(13,17,18,16)(14,22,23,26)(15,25,19,27)(20,28,24,21);(5,9,10,8)(6,11,7,12)(13,15,18,19)(14,20,23,24)(16,25,17,27)(21,26,28,22)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296193);