C₂⁴×#547 #72291283

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂⁴×#547

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

880517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁴×#3496, C₂⁴×D₄×#41
maximal normal subgroup of	C₂⁴×#3496, C₂⁴×D₄×#41
maximal quotient of	C₂⁴×#2407, C₂⁴×#2339, C₂⁴×#2430

GAP

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

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

C24×#547 #72291283