#208962

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116158);