#6399561

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	C₂×#3268	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2329 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,21)(17,22)(18,19)(20,24)(25,27)(26,28);(6,9)(7,11)(13,15,17,18)(14,19,22,23)(16,24,26,25)(20,21,27,28);(13,16)(14,20)(15,24)(17,26)(18,25)(19,21)(22,27)(23,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,26)(18,24)(19,20)(22,28)(23,27);(1,3,4,2)(6,9)(10,12)(13,15,17,18)(14,19,22,23)(16,24,26,25)(20,21,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=fi,e^2=hi,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, 6306757);