C₂²×#2760 #7633558

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×#2760

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂²×#392	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

11026 conjugacy classes of subgroups; only the 3314 normal subgroups are shown.

normal subgroups only

This group has 11026 conjugacy classes of subgroups — too many to draw. Only its 3314 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)(7,11)(13,14,17,15)(16,19,20,18)(21,22)(23,27)(24,25)(26,28);(6,9)(10,12)(14,15)(16,20)(22,25)(23,28);(5,6,8,9)(7,10,11,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,22)(23,26)(24,25)(27,28)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72279216
maximal subgroup of	#72279216, C₂²×#45013, C₂²×#57958, …
maximal normal subgroup of	#72279216, C₂²×#45013, C₂²×#57958, …
maximal quotient of	C₂²×#14572, C₂²×#16357, C₂²×#15369, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540754);