C₂×#7849 #180848

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂×#7849

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂⁴	C₂×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2);(3,4)(5,6)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(4,6)(7,8,9,11,10,12,13,14)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(17,23)(20,26)(24,29)(27,30)

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321583, #72321588, C₂×#193573, …
maximal normal subgroup of	#72321583, #72321588, C₂×#193573, …
maximal quotient of	C₂×#126574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88044);

C2×#7849 #180848

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7849 #180848