C₂³×#1368 #72231990

order	1024
nilpotency class	4
isoclinism	isoclinic class of #365
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂³×#1368

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	#106
commutator	C₈	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂³×#12067, C₂×#72224970, C₂×#72225020, …
maximal normal subgroup of	C₂³×#12067, C₂×#72224970, C₂×#72225020, …
maximal quotient of	C₂×#72231934, C₂×#72224974, C₂×D₄×#1368, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C23×#1368 #72231990