C₂×#33999 #7625211

order	512
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#33999

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂	C₂⁴×C₄²
Frattini, cosocle	C₂³	C₂⁶
socle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

13866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#7629697, C₂×#7625627, C₂×#7629700, …
maximal normal subgroup of	C₂×#7629697, C₂×#7625627, C₂×#7629700, …
maximal quotient of	C₂×D₄×#376, C₂×#513485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532407);

C2×#33999 #7625211