C₂³×#368 #7622538

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂³×#368

Subgroups

	subgroup	G/H
center	C₂³×C₄	D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂⁴	#107
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1986 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 1986 conjugacy classes of subgroups — too many to draw. Only its 1346 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 38 points:

(1,2);(3,4);(5,6);(7,8,11,16,13,17,12,27)(9,18,21,33,23,34,22,37)(10,24,25,31,26,38,19,15)(14,28,29,36,30,35,32,20);(7,9,19,35,12,22,26,36,13,23,25,28,11,21,10,20)(8,14,24,33,16,29,31,34,17,30,38,37,27,32,15,18)

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#2179, C₂³×#1419, C₂³×#1426
maximal normal subgroup of	C₂³×#2179, C₂³×#1419, C₂³×#1426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529734);