C₂×#129555 #72287488

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#129555

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#102
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#3169	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287502, #72287533, C₂×#72225053, …
maximal normal subgroup of	#72287502, #72287533, C₂×#72225053, …
maximal quotient of	C₂×#72225040, C₂×#72225035, #72322611

GAP

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

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

C2×#129555 #72287488