C₂⁴×#2177 #72287463

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂⁴×#2177

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂×D₈
commutator	C₈	C₂⁶×C₄
Frattini, cosocle	C₂×C₈	C₂⁷
prime radical	C₂⁵×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ C₂⁴×C₄² ◁ C₂⁴×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

208712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#12067, #72287579, #72287607
maximal normal subgroup of	C₂⁴×#12067, #72287579, #72287607

GAP

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

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

C24×#2177 #72287463