C₂²×#14860 #72334497

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂²×#14860

Subgroups

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

74733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1259666, C₂²×#613868, C₂×#72334398, …
maximal normal subgroup of	C₂²×#1259666, C₂²×#613868, C₂×#72334398, …
maximal quotient of	C₂×#72334407

GAP

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

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

C22×#14860 #72334497