C₂²×#2835 #7633357

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂²×#2835

Subgroups

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

7178 conjugacy classes of subgroups; only the 3234 normal subgroups are shown.

normal subgroups only

This group has 7178 conjugacy classes of subgroups — too many to draw. Only its 3234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#42364, C₂²×#42548, C₂²×#56441, …
maximal normal subgroup of	C₂²×#42364, C₂²×#42548, C₂²×#56441, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540553);