C₂×#153995 #72294953

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹×C₄²

Aliases

direct product

C₂×#153995

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂³×C₄²
Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

49473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293932, C₂×#72293964, C₂×#72293980, …
maximal normal subgroup of	C₂×#72293932, C₂×#72293964, C₂×#72293980, …

GAP

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

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

C2×#153995 #72294953