C₂²×#2380 #7625927

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵

Aliases

direct product

C₂²×#2380

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂²	C₂⁵×C₄
Frattini, cosocle	C₂³	C₂⁶
prime radical	C₂³×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

14802 conjugacy classes of subgroups; only the 5634 normal subgroups are shown.

normal subgroups only

This group has 14802 conjugacy classes of subgroups — too many to draw. Only its 5634 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) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂²×#2380⋊C₃, #72260711, #72261420, …
maximal subgroup of	C₂×#5020974, C₂×#5021372, C₂×#574938, …
maximal normal subgroup of	C₂×#5020974, C₂×#5021372, C₂×#574938, …
maximal quotient of	C₂×#6407636, C₂×D₃×#2380, C₂³×#2380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533123);