C₂×#8305 #184755

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#8305

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

4098 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 4098 conjugacy classes of subgroups — too many to draw. Only its 418 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) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 30 points:

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

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, 91951);