C₂⁴×#1368 #72287465

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₈ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂⁴×#1368

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁶	#106
commutator	C₈	C₂⁵×C₈
Frattini, cosocle	C₄×C₈	C₂⁶
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

30088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 2048

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#12067, #72287618, #72287624
maximal normal subgroup of	C₂⁴×#12067, #72287618, #72287624

GAP

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

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

C24×#1368 #72287465