C₂×#579825 #72279231

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#579825

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

38789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277929
maximal normal subgroup of	C₂×#72277929
maximal quotient of	C₂×#72277984

GAP

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

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

C2×#579825 #72279231