C₂×#7841 #180784

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₈ · Schur multiplier (H₂) = C₂⁴×C₄×C₈

Aliases

direct product

C₂×#7841

Subgroups

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

1442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#96658, C₂×#128207, C₂×#128226, …
maximal normal subgroup of	C₂×#96658, C₂×#128207, C₂×#128226, …
maximal quotient of	C₂×#94852, C₂×#126531, C₂×C₈×#322

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87980);

C2×#7841 #180784