C₂³×#2112 #72307887

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₈ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂³×#2112

Subgroups

	subgroup	G/H
center	C₂⁴×C₈	C₂³
commutator	C₂²	C₂⁵×C₈
Frattini, cosocle	C₂×C₈	C₂⁶
socle	C₂⁵	C₂²×C₈
prime radical	C₂⁷	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

50933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

448 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂³×#13001, C₂³×#12929, C₂³×#33517, …
maximal normal subgroup of	C₂³×#13001, C₂³×#12929, C₂³×#33517, …

GAP

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

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

C23×#2112 #72307887