C₂×#612424 #72232073

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#612424

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

462365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72220575, C₂×#72222918
maximal subgroup of	#72232191, C₂×#72210469, #72232371, …
maximal normal subgroup of	#72232191, C₂×#72210469, #72232371, …
maximal quotient of	C₂×#72214173, #72236027, C₂×#72210511, …
automorphism group of	#1482, #1483, #1494, …

GAP

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

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

C2×#612424 #72232073