C₂²×#2842 #7633359

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#2842

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂⁷	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13754 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.

normal subgroups only

This group has 13754 conjugacy classes of subgroups — too many to draw. Only its 3490 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) = 128

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	A₄×#2842⋊C₃
maximal subgroup of	#72222731, C₂²×#2842⋊C₃, A₄×#2842, …
maximal normal subgroup of	#72222731, C₂²×#2842⋊C₃, A₄×#2842, …
maximal quotient of	C₂²×#14673, C₂³×#2842, C₂²×#16298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540555);