#392532

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 conjugacy classes of subgroups; only the 386 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 386 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(21,22)(23,29)(24,32)(25,27)(26,30)(28,33)(31,35)(34,36);(6,10)(11,12)(13,15,16,19)(14,17,18,20);(5,7,9,8)(6,11,10,12)(21,23,24,30,25,31,33,36)(22,26,28,29,27,34,32,35);(5,8,9,7)(6,12,10,11)(14,18)(17,20)(22,27)(26,34)(28,32)(29,35)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299728);