C₂⁴×#1333 #72287464

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₈ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂⁴×#1333

Subgroups

	subgroup	G/H
center, socle	C₂⁶	#102
commutator	C₈	C₂⁵×C₈
Frattini, cosocle	C₄×C₈	C₂⁶
prime radical	C₂⁵×D₈	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

97608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 6 generators on 32 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#12067
maximal normal subgroup of	C₂⁴×#12067

GAP

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

gap> G := Group((1,2), (3,4), (9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,30)(21,23)(24,28)(25,27)(26,32)(29,31), (9,11,12,15,13,14,16,10)(18,22,23,31)(19,26)(20,27)(24,30,32,25)(28,29), (5,6), (7,8));

C24×#1333 #72287464